Answer
Symmetric about the origin.
Not symmetric about the y-axis.
Not symmetric about the x-axis.
Work Step by Step
Testing for symmetry:
- about the x-axis: $(r,\theta)$ on the graph $\Rightarrow (r, -\theta)$ or $(-r, \pi-\theta)$ on the graph.
- about the y-axis: $(r,\theta)$ on the graph $\Rightarrow (r, \pi-\theta)$ or $(-r, -\theta)$ on the graph.
- about the origin: $(r,\theta)$ on the graph $\Rightarrow (-r, \theta)$ or $(r, \theta+\pi)$ on the graph.
If $(r,\theta)$ is on the graph, $(-r)^{2}=r^{2}=4\sin(2\theta)$, so
$(-r, \theta)$ is on the graph.
The graph is symmetric about the origin.
Because sine is an odd function,
$4\sin[2(-\theta)]=4\sin(-2\theta)=-4\sin(2\theta)=-(r)^{2}\neq r^{2}$, and
$4\sin[2(\pi-\theta)]=4\sin(2\pi-2\theta)=-4\sin(2\theta)=-(r)^{2}\neq r^{2}$,
Neither $(r, -\theta)$ or $(-r, \pi-\theta)$ are on the graph, so there is no symmetry about x.
There is no symmetry about y, then, either.
To graph, select a few values for $\theta \in[0,\pi/2]$ and calculate $r=2\sqrt{\sin 2\theta}.$
Use the symmetry.
See the resulting image.
